As a function of $n$, what is the set of possible determinants of $n \times n$ matrices whose elements are 0s and 1s and have at most three 1s in each row and column?
I really enjoyed the problem posed on math.stackexchange: https://math.stackexchange.com/questions/4285336/is-there-some-sort-of-property-which-helps-to-get-a-recursion-or-pattern-in-find asking what the possible determinants are of matrices whose elements are 0s and 1s and have at most two 1s in each row and column.
Using the sum over permutations definition of matrix determinant, one is trying to add up the determinants of permutation matrices that can be obtained from the original matrix by turning some 1s to 0s (the other terms of the sum are 0s). By considering "chains" of 1s in the original matrix under the relations "in the same row as" and "in the same column as", one can perform this sum in a systematic way.
I'm curious about the case where there are at most three 1s in each row and column. The structure of 1s under "in the same row as" and "in the same column as" becomes much more complicated, allowing branching and loops.
